1. The problem is to simplify the expression $ \frac{4}{\sqrt{5} - \sqrt{2} - \sqrt{6}} $.\n\n2. First, identify the denominator: $ \sqrt{5} - \sqrt{2} - \sqrt{6} $. The goal is to rationalize this denominator to simplify the expression.\n\n3. Group terms to help with rationalization: consider $ (\sqrt{5} - \sqrt{2}) - \sqrt{6} $.\n\n4. Rationalizing with three terms is tricky, so look for a conjugate. We use the conjugate: $ \sqrt{5} - \sqrt{2} + \sqrt{6} $ to multiply numerator and denominator.\n\n5. Multiply numerator and denominator by $ \sqrt{5} - \sqrt{2} + \sqrt{6} $:\n$$\frac{4}{\sqrt{5} - \sqrt{2} - \sqrt{6}} \times \frac{\sqrt{5} - \sqrt{2} + \sqrt{6}}{\sqrt{5} - \sqrt{2} + \sqrt{6}} = \frac{4(\sqrt{5} - \sqrt{2} + \sqrt{6})}{(\sqrt{5} - \sqrt{2})^2 - (\sqrt{6})^2}$$\n\n6. Calculate the denominator:\n$$ (\sqrt{5} - \sqrt{2})^2 - (\sqrt{6})^2 = (5 - 2\sqrt{10} + 2) - 6 = (7 - 2\sqrt{10}) - 6 = 1 - 2\sqrt{10} $$\n\n7. So the expression is now:\n$$ \frac{4(\sqrt{5} - \sqrt{2} + \sqrt{6})}{1 - 2\sqrt{10}} $$\n\n8. To rationalize the denominator again, multiply numerator and denominator by the conjugate $ 1 + 2\sqrt{10} $:\n$$ \frac{4(\sqrt{5} - \sqrt{2} + \sqrt{6})(1 + 2\sqrt{10})}{(1 - 2\sqrt{10})(1 + 2\sqrt{10})} $$\n\n9. The denominator simplifies to:\n$$ 1 - (2\sqrt{10})^2 = 1 - 4 \times 10 = 1 - 40 = -39 $$\n\n10. So we have:\n$$ \frac{4(\sqrt{5} - \sqrt{2} + \sqrt{6})(1 + 2\sqrt{10})}{-39} = -\frac{4}{39} (\sqrt{5} - \sqrt{2} + \sqrt{6})(1 + 2\sqrt{10}) $$\n\n11. Expand the numerator terms:\n$$ (\sqrt{5} - \sqrt{2} + \sqrt{6})(1 + 2\sqrt{10}) = \sqrt{5} + 2\sqrt{5 \times 10} - \sqrt{2} - 2\sqrt{2 \times 10} + \sqrt{6} + 2\sqrt{6 \times 10} $$\n\n12. Simplify under the radicals:\n$$ = \sqrt{5} + 2\sqrt{50} - \sqrt{2} - 2\sqrt{20} + \sqrt{6} + 2\sqrt{60} $$\n\n13. Simplify $ \sqrt{50} = 5\sqrt{2} $, $ \sqrt{20} = 2\sqrt{5} $, and $ \sqrt{60} = 2\sqrt{15} $:\n$$ = \sqrt{5} + 2 \times 5\sqrt{2} - \sqrt{2} - 2 \times 2\sqrt{5} + \sqrt{6} + 2 \times 2\sqrt{15} $$\n$$ = \sqrt{5} + 10\sqrt{2} - \sqrt{2} - 4\sqrt{5} + \sqrt{6} + 4\sqrt{15} $$\n\n14. Combine like terms:\n$$ (\sqrt{5} - 4\sqrt{5}) + (10\sqrt{2} - \sqrt{2}) + \sqrt{6} + 4\sqrt{15} = -3\sqrt{5} + 9\sqrt{2} + \sqrt{6} + 4\sqrt{15} $$\n\n15. So the entire expression is:\n$$ -\frac{4}{39} \left(-3\sqrt{5} + 9\sqrt{2} + \sqrt{6} + 4\sqrt{15} \right) = \frac{4}{39} \left(3\sqrt{5} - 9\sqrt{2} - \sqrt{6} - 4\sqrt{15} \right) $$\n\n16. This is the simplified form of the original expression.\n\n**Final answer:**\n$$ \frac{4}{39} \left(3\sqrt{5} - 9\sqrt{2} - \sqrt{6} - 4\sqrt{15} \right) $$